Many prior art migration and inversion methods belong to the class of adjoint state problems where forward and backward propagated wavefields are correlated to obtain an image. Examples of such methods include reverse time migration and waveform inversion. These methods require that forward propagated wavefields be accessed in reverse order, in lockstep with the adjoint, backward-propagated wavefields at each time step.
This requirement of simultaneous availability of both the forward and backward-propagated wavefields at each time step poses significant computational challenges for large datasets. Calculating the source and receiver wavefields at every point in the subsurface for every time step has huge requirements in both computational operations and computer memory. Prior art has addressed this problem by reducing the wavefield computations according to an expanding box, wherein for each time step, the wavefield is only calculated for subsurface points that contain the expanding wavefield. At subsurface points outside the expanding box, the wavefield is known to be zero, so there is no need to calculate it. This methodology can save up to 30% of the computational cost of reverse time migration. However, even when using an expanding box for the wavefield propagation, the computational cost of methods such as reverse time migration and waveform inversion is very high.
An expanding box for source wavefield propagation is illustrated by FIG. 1. Panel 10 shows a source wavefield that is being propagated. The majority of the panel is dark gray, indicating that the source wavefield is not calculated in that region and that it is presumed to be zero. The source wavefield is only calculated in the light gray region, which is slightly larger than the actual wavefield, thereby ensuring that the entire wavefield is computed. The propagating wavefield in panel 10 is the half-circle centered at the top of the panel. As the wavefield forward propagates to later times in panels 12 and 14, the light gray region expands to allow the wavefield to be calculated within this expanding box and the dark gray region where the wavefield is held to be zero decreases in size. Due to the complexity of the earth model that the wavefield is being propagated through, the expanding box expands at different rates in different directions and the wavefield becomes complex. The backward propagated receiver wavefield can be similarly calculated within an expanding box. Although the expanding box for source and receiver wavefield propagation can reduce the computational cost by up to 30%, seismic imaging methods such as reverse time migration are still computationally expensive.
There exists a need to process wavefields and to generate images of a subsurface region of interest in a more efficient computational manner.